2013 AMC 12A Problem 2

A softball team played ten games, scoring 1,2,3,4,5,6,7,8,9, and 10 runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

\textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.